The real roots of xˣ = y: every fraction bracket an AI model proposed, decided exactly

x = ln 64 / W(ln 64), where W is Lambert’s function. That form describes the number; it decides nothing. The decimals below are decided by two fractions.

xˣ grows for x above 1, so x is above a fraction p/q exactly when (p/q)^p/q is below 64, that is, when p^p is below 64^q·q^p. For the fraction below, each side of that comparison is a whole number of 2,598,236 digits. The recorder computes both exactly and compares them; the step from x to the comparison is read, not decided (§4).

The two fractions are neighbouring convergents of the root’s continued fraction, [3; 2, 1, 1, 44, 1, 14, 4, 1, 6, …]: the closest fractions for their size, on opposite sides of x. Its large fifth quotient is why 775/228, with a denominator below a thousand, is already off by only 1.266965e-06.

The first two rows are the AI’s four-decimal proposal; the planted rows move it one unit in the last decimal, the wrong way, and are refuted. A refutation here is a comparison of two numbers, so it has no witness point to draw.

Each dot is the root of xˣ = y. A solid dot is trapped between fractions decided here; a hollow dashed one is computed only. The large dots are the roots that are whole numbers.

How tight each trap is

For y up to 100, the bar is the number of decimal places on which the two decided fractions agree: those decimals of the root are decided. The dashed line is how many decimals the closest fraction with a denominator of at most 1,000 shares with the computed root. Where the root is a whole number, the bar reaches the top line, labelled exact.

Every continued fraction, at a glance

Each column is one root, for y from 2 to 400; each row is a partial quotient, the first at the top. The brighter a cell, the larger the quotient, and a large quotient means the convergent before it is an unusually good fraction. Near 27 and 256 the bright cells pile up at the top: there, the whole number is the fraction.

The roots, in detail

The root, the closest fraction and its error are the proposer’s, computed. The bracket is decided here, both sides; the first word is this page’s and the second claimbox’s record, each the same on both sides.

Read from the computed roots. The brackets are decided; the patterns below are observations on computed numbers, and each is labelled as one. The first is a theorem, cited.

One: only kᵏ has a fraction for a root (a theorem, cited)

The root is a whole number exactly at y = 4, 27, 256 and 3,125, and each is pinned from both sides: x ≥ k and x ≤ k, both certified here. For every other whole y the root is not even algebraic. If it were a fraction p/q in lowest terms with q above 1, then p^p = y^q·q^p, which q cannot divide; if it were an irrational algebraic number, xˣ would be transcendental by the Gelfond–Schneider theorem. That theorem is cited, not decided here. So 4 roots are fractions, and every other one is a number no fraction will ever equal.

Two: near kᵏ, one enormous step (an observation)

Just off kᵏ the root sits just off k, by about 1/(kᵏ(1 + ln k)), the reciprocal of the slope of xˣ at k. Its continued fraction says so with one huge quotient: at y = 255 it is 608, where the slope predicts 610.9; at y = 3,126 it is 8,155, against 8154.5. The closest fraction there is the whole number k.

Three: elsewhere, the roots look like typical numbers (an observation)

Pool the partial quotients of all the irrational roots and they follow the law that almost every real number obeys: a quotient of 1 occurs 0.41162 of the time, against the law’s 0.41504. Their geometric mean is 2.6729, against Khinchin’s constant 2.6855, and (ln q_n)/n averages 1.1777, against Lévy’s 1.1866. These are observations on 1,002 computed roots, not a proof.

Four: a few roots have fractions that are almost too good (an observation)

A large partial quotient makes the convergent just before it unusually close. The largest found is 4,651,670, in the root for y = 709. Its convergent sits closer to the root than a fraction of its size usually can. For y = 100, the fraction 795/221 is already off by only 4.433289e-08.